Question

The electric field produced by, a disk of radius R that carries uniformly distributed positive charge q, at a point P located on the axis of the disk (which we choose as the positive z- axis) a distance z from its center is given by equation;

E_z= 1/(4πε_0 ) 2qz/R^2 (1- z/√(z^2+R^2 ))
Starting with above equation write an equation in vector form that gives the electric field when point P is located either on the positive or negative axis of the disk of charge.

Answer 1

Consider our equation:

$$E_z= \frac{1}{4πε_0} \frac{2qz}{R^2} \bigg(1- \frac{z}{\sqrt{z^2+R^2}}\bigg).$$

When we consider the field at very small distances to the disk, in vector form we have

$$E=\frac{q\hat{\textbf{z}}}{4\pi\epsilon_0z^2}\frac{z}{|z|}.$$

The term z/|z| means that the field reverses direction as z changes sign.